Question

A wave equation is represented as $ r=A\sin \alpha \dfrac{\left( x-y \right)}{2}\cos \omega t+\alpha \dfrac{\left( x+y \right)}{2} $ , where $ x $ and $ y $ are in meters and $ t $ is in seconds . Which of the following options is correct?
(A)The wave is a stationary wave
(B)The wave is a progressive wave propagating along positive $ x $ - axis
(C)The wave is progressive propagating at right angles to the positive x-axis
(D)All point lying on line $ y=x+ \dfrac{4\pi }{\alpha } $ are always at rest

Answer 1

Hint :In order to solve this question, first of all we must determine the nature of the given wave, whether it is a standing wave or a progressive wave , that can be determined by comparing the given equation to the general equation for standing and the progressive waves.
General Equation for standing wave: $ y=2A\sin \omega t\cos kx $
General Equation for progressive wave: $ y=A\sin \left( \omega t-kx \right) $
 $ \sin \left( 2n+1 \right)\pi =0 $
 $ v=\dfrac{dr}{dt} $
Where, $ v $ is velocity.

Complete Step By Step Answer:
Firstly, the given wave equation is:
 $ r=A\sin \alpha \dfrac{\left( x-y \right)}{2}\cos \omega t+\alpha \dfrac{\left( x+y \right)}{2} $
The general equations for the standing and the progressive waves are:
For standing wave: $ y=2A\sin \omega t\cos kx $
For progressive wave: $ y=A\sin \left( \omega t-kx \right) $
Since the given wave equation does not match with any of the above equations, it is neither a standing nor a progressive wave
Replace $ y $ by $ y=x+\dfrac{4\pi }{\alpha } $
 $ \begin{align}
  & r=A\sin \left[ \alpha \left( \dfrac{x-x-\dfrac{4\pi }{\alpha }}{2} \right) \right]\cos \left[ \omega t-\alpha \left( \dfrac{x+x+\dfrac{4\pi }{\alpha }}{2} \right) \right] \\
 & \Rightarrow r=A\sin \left( -2\pi \right)\cos \left( \omega t-\alpha x-2\pi \right) \\
 & \Rightarrow r=0 \\
\end{align} $
And finding the velocity, we get
 $ \dfrac{dr}{dt}=0\Rightarrow v=0 $
Therefore, we can say that all points lying at the line with the equation $ y=x+\dfrac{4\pi }{\alpha } $ are always at rest because the velocity is zero.
Therefore, option (D) is correct.

Note :
According to the wave equation, this wave is neither a standing nor a progressive wave . This rules out the first three options and then $ y $ is replaced by $ y=x+\dfrac{4\pi }{\alpha } $ to get a result matching with fourth. But to rule out other options is equally important.